(a) Consider the 1-forms wi = ydx + xdy + zdz and w2 = xda + zdy + ydz in R3. Compute their wedge product, w1 A w2. Note that w1 A w2 is a 2-form on R.", so you want to put it in the form Wi A w2 = A(x, y, z) dy A dz + B(x, y, z) dz A dx + C(x, y, z) dx A dy. (b) Now recall that 1-forms on R." are represented by vector fields. For example, w1 is represented by the vector field F = (y, x, z), and w2 is represented by the vector field G = (x, z, y). Compute the cross product F X G. (c) Finally, recall that 2-forms on Rare also represented by vector fields: we say that the 2- form A dy A dz + Bdz A da + C dx A dy is represented by the vector field (A, B, C'). Now compare your answers in (a) and (b), and fill in the blanks (with the name of the suitable operation):. upon identifying 1-forms and 2-forms on R. with vector fields, we see that the product of 1-forms is represented by the product of vector fields. (Note that for this identification to be correct, we really need to write the 2-form in that specific form with dy A dz, dz A dx and da A dy!) (d) For the final activity, let's just make sure you know how the exterior derivative works. In class, we learned how to differentiate a 1-form to onbtain a 2-form. We can mimic that definition so that we can now differentiate a 2-form and obtain a 3-form! Compute dw when w is the 2-form W = 134 dx1 / dx3 + 2x1+2+23-24 dx2 Adx4.Note that dw is a 3-form in R*, so your answer should looks like dw = Adx1 A dx2 A dx3 + Bdx1 A dx2 A dx4 + Cdx1 A dx3 A dx4 + Ddx2 A dx3 for suitable functions A, B, C and D (some of them possibly equal to 0)